As usual, the analysis of the situation is excellent. I tend to disagree with some of Cannoli's conclusions, or the absolute extent of them (though backing what you say is important, of course).
In short, a cooperation/collaboration/unification between Rand and White Tower is not a "magic bullet" for the fight against the DO & Forsaken. I would add that Rand's alliance with the scummy, slaving Seanchan is far more important. The reasons for the WT's "necessary" support are largely irrelevant or specious at best - particularly 1 and 2, the ones the AS have always stressed, and 4a (which the fans have oft maintained & Egwene claimed to Rand's face once IIRC).
Still, Min's vision from book 1 on, and implications from RJ's approach to the Creator vs DO struggle, say that he really should have them on his side. The fireflies hold their own against the great void when they are stronger, united, and can bring more numbers together. The One Power always made the best works when male and female halves were used together.
Basically, the good guys gotta work together to beat the Shadow. Everyone who believes that the DO musn't win, be they Tinker, Whitecloak, or even lousy, foolish Seanchan, should rise against the utter doom that they all face. The fights in Emond's Field (I know the WC didn't actually help, but everyone else did) and at The Cleansing were signs of this. For the latter, it's laid out that the Shadow loses because the other group were willing to (a) work together and (b) give up control to circles. They held back the 7? of the strongest, most-experienced and talented channelers alive. I was annoyed by the tally at the end, til I realized they were actually lucky (RJ flat-out said it).
The Dark competes among themselves brutally and ruthlessly, and this somewhat hampers what they can accomplish. The Light can cohere better - with many ensouled people, no less (unlike Trollocs & such) - and so they are stronger.
Given the odds that Rand must face, he can't afford to turn people away. It won't be remotely practical since there will simply be too many Trollocs (it's too quiet up there) and Dreadlords (thanks partly to Rand's carelessness). At the least, the WT (if they don't act like self-indulgent & mean children) will not interfere with him, e.g., two separate but simultaneous attacks on a Dark stronghold. At the best, they might be able to beat a lot of bad guys off handily.
Perhaps experience guiding circles is the best thing the WT can offer, besides trained numbers.
I guess the WT is like almost every major institution in the series: they are old and comfortable and prone to over-estimating themselves while under-estimating others. One king of Cairhien thought he could make a nifty wood throne, and another thought killing a gleeman was no problem - they both died badly. Not only were none of them prepared to deal with the Dragon or the changes he would bring, none of them were nearly as strong or invulnerable as they believed (Aiel possibly excepted). Perhaps RJ was very intent on making this point - and the WT is an analogue for the Catholic Church (the Rome-governed organizational part of it). The WT matches its real-life counterpart for arrogance and adherence to certain antiquated mentalities that are neither beneficial nor helpful to its supposed ultimate goal ("evil gets whupped".
The White Tower is full of a ton of self-important people who must unlearn many of the traditions that they blindly accepted as a way of making themselves feel more important. Still, when a house in the center of your town is burning, you'll see people who hate each other pass off buckets of water to make sure that everyone is safe.